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Properties

Label 18.0.32204753206...9648.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 11^{9}\cdot 37^{15}$
Root discriminant $106.71$
Ramified primes $2, 11, 37$
Class number $110592$ (GRH)
Class group $[2, 2, 8, 24, 144]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92027224, -86126792, 118790560, -46835020, 67169500, -32206984, 19106088, -10931026, 4599882, -2078441, 732082, -203491, 66249, -14219, 2176, -382, 65, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 65*x^16 - 382*x^15 + 2176*x^14 - 14219*x^13 + 66249*x^12 - 203491*x^11 + 732082*x^10 - 2078441*x^9 + 4599882*x^8 - 10931026*x^7 + 19106088*x^6 - 32206984*x^5 + 67169500*x^4 - 46835020*x^3 + 118790560*x^2 - 86126792*x + 92027224)
 
gp: K = bnfinit(x^18 - x^17 + 65*x^16 - 382*x^15 + 2176*x^14 - 14219*x^13 + 66249*x^12 - 203491*x^11 + 732082*x^10 - 2078441*x^9 + 4599882*x^8 - 10931026*x^7 + 19106088*x^6 - 32206984*x^5 + 67169500*x^4 - 46835020*x^3 + 118790560*x^2 - 86126792*x + 92027224, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 65 x^{16} - 382 x^{15} + 2176 x^{14} - 14219 x^{13} + 66249 x^{12} - 203491 x^{11} + 732082 x^{10} - 2078441 x^{9} + 4599882 x^{8} - 10931026 x^{7} + 19106088 x^{6} - 32206984 x^{5} + 67169500 x^{4} - 46835020 x^{3} + 118790560 x^{2} - 86126792 x + 92027224 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3220475320686797629881417736272539648=-\,2^{12}\cdot 11^{9}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.71$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{36} a^{15} - \frac{1}{4} a^{14} + \frac{1}{36} a^{13} + \frac{1}{18} a^{12} + \frac{2}{9} a^{11} - \frac{5}{12} a^{10} + \frac{5}{36} a^{9} + \frac{5}{36} a^{8} + \frac{5}{18} a^{7} - \frac{5}{36} a^{6} + \frac{5}{18} a^{5} - \frac{1}{18} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{36} a^{16} - \frac{2}{9} a^{14} - \frac{7}{36} a^{13} + \frac{2}{9} a^{12} + \frac{1}{12} a^{11} + \frac{7}{18} a^{10} + \frac{7}{18} a^{9} + \frac{1}{36} a^{8} - \frac{5}{36} a^{7} - \frac{17}{36} a^{6} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a$, $\frac{1}{57257605577587897130156515123613299602720198714701748818137853631962916} a^{17} + \frac{462236161907112741443410122921964890025049398802112233298095501657231}{57257605577587897130156515123613299602720198714701748818137853631962916} a^{16} + \frac{90340311840012006383740987903755551274676401441486769113500229169549}{14314401394396974282539128780903324900680049678675437204534463407990729} a^{15} - \frac{2729216967388269740564829373563887133217132288117439348054781388896329}{57257605577587897130156515123613299602720198714701748818137853631962916} a^{14} - \frac{1293012245599559779196568793899206338675163027828754864728590174277479}{19085868525862632376718838374537766534240066238233916272712617877320972} a^{13} + \frac{3779860466437435997892771932184046472005610584568055647212560686495779}{57257605577587897130156515123613299602720198714701748818137853631962916} a^{12} - \frac{16237758931961508272374831010261027442612010922120527086748728695426167}{57257605577587897130156515123613299602720198714701748818137853631962916} a^{11} + \frac{1852582283184335994182249124310035982479662313397741432666677647734093}{4771467131465658094179709593634441633560016559558479068178154469330243} a^{10} - \frac{4802505879123567176153671003510942072723237528588851754164192575701087}{57257605577587897130156515123613299602720198714701748818137853631962916} a^{9} - \frac{477372601444822876590632440969283779269356797000220651546382153491557}{9542934262931316188359419187268883267120033119116958136356308938660486} a^{8} - \frac{366244412583790103120555240510729551980409621569240547237228593709321}{1590489043821886031393236531211480544520005519852826356059384823110081} a^{7} + \frac{19060299491008587276296682356577308850299184181367922697786468008521}{147952469192733584315649909880137725071628420451425707540407890521868} a^{6} - \frac{12909118239976827886435813098405328750722869487898117964827804703120755}{28628802788793948565078257561806649801360099357350874409068926815981458} a^{5} + \frac{833289829485173681457115894663312666947456015912454683444375641912973}{3180978087643772062786473062422961089040011039705652712118769646220162} a^{4} - \frac{2279093356753688091551777580119736820362697227816557793201755583172886}{14314401394396974282539128780903324900680049678675437204534463407990729} a^{3} + \frac{1767250415319154907520938363076543889844768660083470190690362056818285}{14314401394396974282539128780903324900680049678675437204534463407990729} a^{2} + \frac{3561312414473649463757232658173291000201410914239822909573842377391489}{14314401394396974282539128780903324900680049678675437204534463407990729} a - \frac{41404964812624849489315214421162278913091488158926883960271377983930}{110964351894550188236737432410103293803721315338569280655305917891401}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{24}\times C_{144}$, which has order $110592$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-407}) \), 3.3.148.1, 3.3.1369.1, 6.0.1078706288.1, 6.0.92296806767.2, 9.9.6075640136512.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$